Completeness on the tangent bundle

Question

I was wondering if geodesics are defined for all time on compact Finsler manifolds, or more generally, for any spray on a compact manifold (where by geodesics, I simply mean the integral curves of the spray).

I thought perhaps that the homogeneity condition of a spray would mean that the "size" of vectors couldn't grow too quickly along integral curves. To give a meaning to "size," I thought it would be useful to endow $TM$ with a complete metric, and the natural choice seemed to be the Sasaki metric. But then I realized, I didn't actually know if compactness of $M$ guarantees that the Sasaki metric is complete (I now know that it is). So I ask the following three sub-questions:

1. If $(M,g)$ is complete, is the Sasaki metric on $TM$ necessarily complete?
2. If $M$ is compact, is every spray on $M$ a complete vector field on $TM$?
3. If $M$ is compact and Finsler, do geodesics on $M$ exist for all time?

Answer 1

These answers are primarily due to Juan Carlos Álvarez-Paiva's comments. I'm just recounting them:

1. If $(M,g)$ is a complete, then $TM$ is also complete under the Sasaki metric, as shown here.

2. Not every spray over a compact manifold is complete. Álvarez-Paiva linked this paper, which gives (as part of a more detailed study) an example of a Lorentzian metric on a 2-torus, where geodesics are not defined for all time. The idea is that a closed geodesic $\gamma:[0,1]\rightarrow M$ may increase in speed: $$\gamma(1)=\gamma(0)\quad\text{and}\quad\lambda\cdot \gamma'(1)=\gamma'(0)\quad\text{with}\quad\lambda<1.$$ Then $\gamma$ extends to go around infinitely many times and on the $n^\text{th}$ time around, the initial speed is $\lambda^{1-n}\gamma'(0)$ and the orbit takes time $\lambda^{n-1}$. Hence, the first $n$ times around takes time $$\sum_{k=0}^{n-1}\lambda^{k}=\frac{1-\lambda^{n}}{1-\lambda}.$$ As $n\rightarrow \infty$, this time converges to a finite number $\frac{1}{1-\lambda}$, which is the furthest that $\gamma$ extends.

3. Again pointed out by Álvarez-Paiva: on a compact Finsler manifold, the sphere bundles are compact and the geodesic spray is tangent to these sphere bundles. Therefore, in this case, geodesic spray is complete!